There are two ways to show that P(X>n) = (1 − p) n. The easiest way is to realize that P(X>n) is the probability that we had “no success in the first n trials,” which clearly equals (1 − p) n. A more...

There are two ways to show that P(X>n) = (1 − p) n. The easiest way is to realize that P(X>n) is the probability that we had “no success in the first n trials,” which clearly equals (1 − p) n. A more involved way is by calculation:

If we recall from calculus that

the answer follows immediately.